<!DOCTYPE HTML>
<html>
<head>
<meta charset="UTF-8">
<title>Section 5.2.4: Solves a simple QCQP</title>
<link rel="canonical" href="/Users/mcgrant/Repos/CVX/examples/cvxbook/Ch05_duality/html/qcqp.html">
<link rel="stylesheet" href="../../../examples.css" type="text/css">
</head>
<body>
<div id="header">
<h1>Section 5.2.4: Solves a simple QCQP</h1>
Jump to:&nbsp;&nbsp;&nbsp;&nbsp;
<a href="#source">Source code</a>&nbsp;&nbsp;&nbsp;&nbsp;
<a href="#output">Text output</a>
&nbsp;&nbsp;&nbsp;&nbsp;
Plots
&nbsp;&nbsp;&nbsp;&nbsp;<a href="../../../index.html">Library index</a>
</div>
<div id="content">
<a id="source"></a>
<pre class="codeinput">
<span class="comment">% Boyd &amp; Vandenberghe, "Convex Optimization"</span>
<span class="comment">% Jo&euml;lle Skaf - 08/23/05</span>
<span class="comment">%</span>
<span class="comment">% Solved a QCQP with 3 inequalities:</span>
<span class="comment">%           minimize    1/2 x'*P0*x + q0'*r + r0</span>
<span class="comment">%               s.t.    1/2 x'*Pi*x + qi'*r + ri &lt;= 0   for i=1,2,3</span>
<span class="comment">% and verifies that strong duality holds.</span>

<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>,13);
n = 6;
P0 = randn(n); P0 = P0'*P0 + eps*eye(n);
P1 = randn(n); P1 = P1'*P1;
P2 = randn(n); P2 = P2'*P2;
P3 = randn(n); P3 = P3'*P3;
q0 = randn(n,1); q1 = randn(n,1); q2 = randn(n,1); q3 = randn(n,1);
r0 = randn(1); r1 = randn(1); r2 = randn(1); r3 = randn(1);

fprintf(1,<span class="string">'Computing the optimal value of the QCQP and its dual... '</span>);

cvx_begin
    variable <span class="string">x(n)</span>
    dual <span class="string">variables</span> <span class="string">lam1</span> <span class="string">lam2</span> <span class="string">lam3</span>
    minimize( 0.5*quad_form(x,P0) + q0'*x + r0 )
    lam1: 0.5*quad_form(x,P1) + q1'*x + r1 &lt;= 0;
    lam2: 0.5*quad_form(x,P2) + q2'*x + r2 &lt;= 0;
    lam3: 0.5*quad_form(x,P3) + q3'*x + r3 &lt;= 0;
cvx_end

obj1 = cvx_optval;
P_lam = P0 + lam1*P1 + lam2*P2 + lam3*P3;
q_lam = q0 + lam1*q1 + lam2*q2 + lam3*q3;
r_lam = r0 + lam1*r1 + lam2*r2 + lam3*r3;
obj2 = -0.5*q_lam'*inv(P_lam)*q_lam + r_lam;

fprintf(1,<span class="string">'Done! \n'</span>);

<span class="comment">% Displaying results</span>
disp(<span class="string">'------------------------------------------------------------------------'</span>);
disp(<span class="string">'The duality gap is equal to '</span>);
disp(obj1-obj2)
</pre>
<a id="output"></a>
<pre class="codeoutput">
Computing the optimal value of the QCQP and its dual...  
Calling Mosek 9.1.9: 35 variables, 10 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 10              
  Cones                  : 4               
  Scalar variables       : 35              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 10              
  Cones                  : 4               
  Scalar variables       : 35              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 10
Optimizer  - Cones                  : 4
Optimizer  - Scalar variables       : 35                conic                  : 32              
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 49                after factor           : 55              
Factor     - dense dim.             : 0                 flops                  : 1.25e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   5.6e+00  5.0e+00  1.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   1.0e+00  0.00  
1   1.8e+00  1.6e+00  3.6e-01  -5.80e-01  9.950141487e-02   7.175750246e-01   3.3e-01  0.01  
2   4.3e-01  3.8e-01  3.8e-02  7.42e-01   -2.392908578e+00  -2.297090534e+00  7.7e-02  0.01  
3   5.3e-02  4.7e-02  1.8e-03  8.74e-01   -3.035369662e+00  -3.018245201e+00  9.5e-03  0.01  
4   1.4e-02  1.2e-02  2.5e-04  9.53e-01   -3.163983578e+00  -3.158419311e+00  2.4e-03  0.01  
5   2.2e-03  2.0e-03  1.9e-05  8.95e-01   -3.217054679e+00  -3.215643247e+00  3.9e-04  0.01  
6   2.0e-05  1.8e-05  1.5e-08  1.00e+00   -3.225854306e+00  -3.225843854e+00  3.6e-06  0.01  
7   2.0e-06  1.8e-06  4.9e-10  1.00e+00   -3.225929324e+00  -3.225928277e+00  3.6e-07  0.01  
8   1.6e-07  1.4e-07  1.1e-11  1.00e+00   -3.225937738e+00  -3.225937655e+00  2.8e-08  0.01  
9   1.9e-08  1.7e-08  4.4e-13  1.00e+00   -3.225938379e+00  -3.225938370e+00  3.3e-09  0.01  
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -3.2259383795e+00   nrm: 7e+00    Viol.  con: 4e-08    var: 0e+00    cones: 0e+00  
  Dual.    obj: -3.2259383698e+00   nrm: 7e-01    Viol.  con: 0e+00    var: 4e-08    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 9         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.895296
 
Done! 
------------------------------------------------------------------------
The duality gap is equal to 
  -5.8546e-08

</pre>
</div>
</body>
</html>